Applications of Physics to Finance and Economics: Returns, Trading Activity and Income

This dissertation reports work where physics methods are appliedto financial and economical problems. Some material in this thesisis based on $3$ published papers \cite{SY,SPY,income} which dividethis study into two parts. The first part studies stock marketdata (chapter 1 to 5). The second part is devoted to personalincome in the USA (chapter 6).We first study the probability distribution of stock returns at mesoscopictime lags (return horizons) ranging from about an hour to about amonth. While at shorter microscopic time lags the distribution haspower-law tails, for mesoscopic times the bulk of the distribution(more than 99\% of the probability) follows an exponential law. Theslope of the exponential function is determined by the variance ofreturns, which increases proportionally to the time lag. At longertimes, the exponential law continuously evolves into Gaussiandistribution. The exponential-to-Gaussian crossover is welldescribed by the analytical solution of the Heston model withstochastic volatility.After characterizing the stock returns at mesoscopic time lags, westudy the subordination hypothesis with one year of intraday data.We verify that the integrated volatility $V_t$ constructed fromthe number of trades process can be used as a subordinator for adriftless Brownian motion. This subordination will be able todescribe $\approx 85\%$ of the stock returns for intraday timelags that start at $\approx 1$ hour but are shorter than one day(upper time limit is restricted by the short data span of oneyear). We also show that the Heston model can be constructed bysubordinating a Brownian motion with the CIR process. Finally, weshow that the CIR process describes well enough the empirical$V_t$ process, such that the corresponding Heston model is able todescribe the log-returns $x_t$ process, with approximately themaximum quality that the subordination allows ($80\% - 85\%$).Finally, we study the time evolution of the personal incomedistribution. We find that the personal income distribution in theUSA has a well-defined two-income-class structure. The majority ofpopulation (97--99\%) belongs to the lower income class characterized by the exponential Boltzmann-Gibbs (``thermal'') distribution, whereas the higher income class (1--3\% of population) has a Pareto power-law (``superthermal'') distribution. By analyzing income data for 1983--2001, we show that the ``thermal'' part is stationary in time, save for a gradual increase of the effective temperature, whereas the ``superthermal'' tail swells and shrinks following the stock market. We discuss the concept of equilibrium inequality in a society, based on the principle of maximal entropy, and quantitatively show that it applies to the majority of population.